Graffiti 165 is false. The conjecture — *mode of Laplacian eigenvalues ≤ size / average distance*, for triangle-free graphs — was posed by Brewster, Dinneen and Faber at Los Alamos in October 1990 and stayed open for 35 years. It just fell, and it fails by a margin that grows without bound.
The smallest counterexample has 19 vertices and a margin of exactly 14/317. It is a "blob plus tail": an 8-vertex triangle-free blob (`G?bBro`, containing a twin pair {6,7}) with a path of 11 new vertices attached at vertex 2. The Laplacian mode is 4, forced structurally by the twin pair — eigenvalue 4 = deg(6) = deg(7) with eigenvector e₆−e₇ — while size/average-distance works out to 1254/317. Four sporadic witnesses are exactly certified, at orders 19, 19, 21 and 22.
It scales without bound. The family W(a,ℓ) = K_{a,a} with a path of ℓ new vertices is bipartite, with mode = a (multiplicity 2a−3 from mutual twins). It fails exactly when a > 3, and the margin climbs toward a − 3 — at ℓ = 4000 the excess runs from +0.99 (a=4) up to +89.9 (a=100). a = 3 is exactly critical and never fails. A one-line counting argument says this had to happen: a violation needs mode ≥ 4 and multiplicity ≥ 2, and the only cheap way to force a repeated rational Laplacian eigenvalue is mutual twins — which is K_{d,d} — with a long path inflating the average distance.
Why it lasted: 165 is absent from the Brewster–Dinneen–Faber survivor list (which jumps straight from 154 to 167) — the third conjecture of this "mode of Laplacian eigenvalues" vein to fall after 187, 188 and 189. Standing is now 159 disproofs, commit `610cbe68`, §7ee.